$A \in M_{m \times 1}(\Bbb C),\ B \in M_{m \times n}(\Bbb C),\ C \in M_{n \times 1}(\Bbb C)$. Now when I do matrix multiplication $A^\dagger BC$ where $A^\dagger$ represents the tranpose conjugate of the vector I get a complex number, say $a+ib$. I want to find the modulus squared of that, i.e. $|A^\dagger BC|^2$ but how do I represent this in terms of matrices?
I have the answer: $A^\dagger BCC^\dagger B^\dagger A$. But how to prove it?
Essentially we want to multiply $A^\dagger BC$ by its complex conjugate. However, the conjugate transpose will also work (all $1\times 1$ matrices are transpose invariant). So what is the conjugate transpose of $A^\dagger BC$?